The Co-universal C-algebra of a Row-finite Graph

Let E be a row-finite directed graph. We prove that there exists a C∗algebra C∗ min (E) with the following co-universal property: given any C∗-algebra B generated by a Toeplitz-Cuntz-Krieger E-family in which all the vertex projections are nonzero, there is a canonical homomorphism from B onto C∗ min (E). We also identify when a homomorphism from B to C∗ min (E) obtained from the co-universal property is injective. When every loop in E has an entrance, C∗ min (E) coincides with the graph C∗-algebra C∗(E), but in general, C∗ min (E) is a quotient of C∗(E). We investigate the properties of C∗ min (E) with emphasis on the utility of co-universality as the defining property of the algebra.